People who are competent at mental arithmetic routinely use the distributive law when doing multiplication. If asked to evaluate $38\cdot14$, I would immediately break it up as $38\cdot10+38\cdot4=380+38\cdot2\cdot2$.
However, I think it's more important to focus on the meaning of the distributive law rather than on its applications. The meaning is of course that if I have $a$ of something in one hand and $b$ in the other, than I have $a+b$ in total, and this applies even if that "something" is $x$ of some object, thus we conclude that $ax+bx=(a+b)x$.
Maybe you're worried that once this is explained, students will feel it's so obvious that it couldn't possibly be of any use. Even in the mental arithmetic example I gave above, you don't have to be explicitly aware of the symbolic law $ax+bx=(a+b)x$ to apply that kind of reasoning, most people do it without thinking. You only really appreciate the value of the law when you do algebra.
I think most students should be able to solve equations of the form $ax=b$ even before taking algebra. Until you've taken algebra, though, an equation like:
seems utterly intractable. What's remarkable is that through simple application of distributivity, the above reduces to the form $ax=b$ and can then immediately be solved.